I've done that - I've allowed them to put my account into "troubleshooting" mode, and reset my iCloud password.
I have an iPhone 6s 64gb on 9.1.
This issue started for me on the 14th November - I accidentally deleted something, so reset my phone and restored to an iCloud backup from just 2 hours earlier. I had the white screen with black apple logo, no progress bar. Force restarted and had all apps' data missing, no health data, no messages, and many apps quit as soon as they're launched. Some launched, but display nothing, others launch and are empty as though I've just re-installed them. So I've had to re-install most apps to be able to use them again.
I spoke with Apple support on the 15th, tried a few things, spoke again over the following few days, when they eventually did the troubleshooting mode on Wednesday 18th, logging it with the Engineering team.
I rang back a few times (the lady, a Senior Technician, gave me her contact details so I could keep speaking to her about it), and there was no feedback from Engineering, other than they had a backlog of cases to work through.
I had a call today, letting me know engineering have looked at it and there's been a few reports of this issue.
Engineering asked her to try restoring my backup to see if it worked for her, but she couldn't get it to complete so she's asking Engineering to try it.
They've asked me to restore a backup from my iPhone 6 (I never deleted those backups from iCloud), which I'm currently doing - that backup is from 24th September. I've also tried earlier backups between then and november 14th, but slow internet so haven't tried many. They didn't work (same issue, missing data).
What I have noticed, the white screen happens when doing the "Erase all content and settings" on the device and then restoring an iCloud backup. The white screen doesn't happen when Restoring the software on iTunes, then restoring an iCloud backup.
Both methods of wiping result in missing data, however, so doesn't make much difference.
I'll see how this 24th September iPhone 6 backup goes (28 minutes remaining..), and I'll post back any info I get back from Apple.
Sorry for the long post - I felt like I was writing a diary!